A note on monogenic even polynomials

2025/05/15 by Joachim König, König, Joachim
Mathematics · #Advanced Mathematical Theories #Algebraic and Geometric Analysis #FOS: Mathematics #Mathematics and Applications #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2505.10119

openalex publication_date 2025/05/15 · openalex created_date 2025/10/15 · openalex updated_date 2026/07/28

Abstract

We extend several predecessor works on even sextic monogenic polynomials. In particular, we prove a conjecture of Lenny Jones, thereby classifying even sextic monogenic polynomials with cyclic Galois group. This result is key to completing previous partial results on existence or non-existence of infinite families of even sextic monogenic polynomials with a prescribed Galois group. Some of the underlying ideas are relevant for investigation of more general families of even polynomials f(X²), or power-compositional polynomials f(X^ℓ).

Citations

• Investigating monogenity in a family of cyclic sextic fields
• Monogenic Even Cyclic Sextic Polynomials
• Monogenic Even Octic Polynomials and Their Galois Groups
• Monogenic cyclotomic compositions
• Integral bases and monogenity of the simplest sextic fields

Related

• Monogenic Even Cyclic Sextic Polynomials
• Investigating monogenity in a family of cyclic sextic fields
• Monogenic Even Octic Polynomials and Their Galois Groups
• Monogenic sextic trinomials x⁶+Ax³+B and their Galois groups
• Monogenic Cyclic Polynomials in Recurrence Sequences